It's always good to see the same problem done more than one
way, and it's especially true with these ratio problems
because they, in general, can be done in many, many different
ways, and different types of ways gel with different people.
So I just did a video on this where I said that the
ratio of apples to oranges starts off at 5 to 8.
And we take away 15.
We take away 15 apples, and the ratio becomes 1 to 4.
And then the question that we have to answer is how much
total fruit do we have after taking away the 15 apples?
And we figured it out.
We figured out that it's 50 pieces of fruit.
But let's see if we can do it a different way.
The last time I kind of talked through it and
used a lot of words.
Let see if we can do it a little bit more algebraically.
So let a is equal to the number of apples we are starting with,
and let o equal the number of oranges.
That doesn't change throughout the whole problem.
So we're starting off with a apples and o oranges.
So the ratio of a to o is equal to what?
The ratio of a-- let me actually-- well, instead of
trying to fit this whole problem into this part of the
screen, let me just copy and paste this part, and do it at
the bottom part of the screen.
Let me copy it, and then let me scroll down here where I have
some extra real estate because I think I'm going to need more
than that little pocket of area right there.
So I copied and pasted just like that.
OK, so there we have it.
a is the number of apples to start with, and o is
the number of oranges.
So we could write-- let me do it in a color that
I haven't used before.
I actually haven't used white.
So we can say that the ratio of apples to oranges
is equal to 5 to 8.
That's what they tell us, that the number of apples we start
with divided by the number of oranges, or the ratio of them--
i have to be a little bit more particular-- is 5 to 8.
Then we removed 15 apples.
So if we remove 15 apples, how many apples do we have?
We have a minus 15 apples.
And they tell us that the ratio-- then the ratio of
apples to oranges becomes 1 to 4.
So a minus 15, which is the number of apples we start with
minus 15, the ratio of that to oranges is equal to 1 over 4.
And let's see if we can solve these two equations.
I have two equations here, and I have two unknowns.
Let's see what we can do here.
And I should probably shouldn't have labeled-- there's a reason
why o isn't used in algebra a lot, because it looks just
like a zero, but I'll try to be careful.
So this first-- let me write down my two
equations over here.
It never hurts to rewrite it.
The original apples to oranges ratio is 5 to 8.
Then when we remove 15 apples, the ratio of our new number of
apples to oranges is equal to 1 over 4.
This is just an algebraic way of stating our problem.
Let's see if we can solve this.
If we cross-multiply both sides of this equation,
we get 8a is equal to 5o.
That's not a zero.
It's equal to 5o.
Or if we subtract 5o from both sides, get 8a
minus 5o is equal to 0.
So this is just a linear equation with two unknowns.
Now, what do we get on this right-hand side?
If we cross-multiply, we get 4a minus 60.
I'm just multiplying 4 times a minus 15.
That is equal to 1 times o, or we could just write--
well, I'll write 1o.
That's not a 10.
That's 1o, or 1 orange.
And if we rearrange this, if we subtract the 1o from both
sides, we get 4a minus 1o minus 60 is equal to 0.
Or if we add 60 to both sides, we get 4a minus
1o is equal to 60.
And here, you might recognize that these are just two
linear equations right here.
That's one of them, and that's the second of them.
And I have two unknowns.
I have a and o.
I probably should have called them x and y to prevent this o
from looking so confusing because that's a zero
and that's an o.
But I think you get what I'm saying.
So how do we solve this?
Well, what we can do is we can multiply one of these times a
convenient factor, and then add them or subtract them to
get rid of one of the two variables, so let's do that.
Let's multiply this equation times minus 2, so it becomes
minus 8a plus 2o is equal to minus 120.
I just multiplied this equation right here times minus 2.
If I do that and then if I add it to this equation right here,
this equation is 8a minus 5o is equal to 0, what do I get?
Let's just add these two equations.
This becomes 0a.
Minus 5 plus 2 is minus 3o is equal to minus 120, or o is
equal to-- you divide both sides-- you can ignore this.
You divide both sides by minus 3, you get o is equal to 40.
So let's see.
So the oranges is equal to 40.
And that's before and after.
The o never changes.
o is equal to 40.
So what are our apples equal to?
We could go to either of these equations.
We could go to any of these that we have.
Maybe we can go to this one.
So we have 8 times our number of apples is equal to 5 times
the number of oranges, or 5 times 40.
8 times the number of apples is equal to 200, or the apples is
equal to 200 over 8, which is equal to 100 over 4,
which is equal to 25.
So remember what a and o are.
O is the number of oranges we have throughout the problem.
a is the number of apples we start with.
That's what we start with.
So you can say that we start with 25 apples, 40 oranges,
or we start with 65 pieces of fruit.
Then if you remove 15 fruit, you'll end up with 50 fruit,
which is the exact same answer we got the other way of doing
this problem, and it's nice to see it done two ways.
If you want to know the number of apples we end up with,
we started with 25 apples.
If you were to take 15 away from them, you're going
to get 10 apples.
So you start with 25 apples, 40 oranges, 65 fruit.
You end with 10 apples, 40 oranges, or 50 fruit.
Either way.
Hopefully, it was reasonably useful to see this
done another way.